Roberts was born in Derby, the son of Edna (Allsop) and John Roberts, an auto mechanic. When he was four, Roberts' family moved to Bath. In Bath, he attended City of Bath Boys' School. As a child he at first wanted to be a detective and then, when given a chemistry set, a Chemist. In 1965 he graduated from the University of Sheffield with a Bachelor of Science degree in Chemistry followed by a PhD in 1969. His thesis involved phytochemical studies of neoflavonoids and isoflavonoids.

During 1969–1972, he did postdoctoral research at Harvard University. before moving to Cold Spring Harbor Laboratory, where he was hired by James Dewey Watson, a co-discoverer of the structure of DNA and a fellow Nobel laureate. In 1992, he moved to New England Biolabs. The following year, he shared a Nobel Prize with his former colleague at Cold Spring Harbor Phillip Allen Sharp.

In 1992, Roberts received an honorary doctorate from the Faculty of Medicine at Uppsala University, Sweden. After becoming a Nobel Laureate in 1993 he was awarded an Honorary Degree (Doctor of Science) by the University of Bath in 1994.

Roberts was elected a Fellow of the Royal Society (FRS) in 1995 and a member of the European Molecular Biology Organization (EMBO) in the same year. In 2005, a multimillion-pound expansion to the chemistry department at the University of Sheffield, where he had been a student, was named after him. A refurbished science department at Beechen Cliff School (previously City of Bath Boys' School) was also named after Roberts, who had donated a substantial sum of his Nobel prize winnings to the school.

Roberts is an atheist and was one of the signers of the Humanist Manifesto. He was knighted in the 2008 Birthday Honours.

Roberts has been a keynote speaker at the Congress of Future Medical Leaders (2014, 2015, 2016).